# A Sequential Algorithm for Signal Segmentation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bayesian Model for Power Switch

## 3. The Sequential Segmentation Algorithm

**seg**($y\in {\Re}^{N}$,${n}_{min}$) as:

- 1
**If**$N<{n}_{min}$, stop, returning the empty vector $t=\left[\right]$;- 2
**Else**- (a)
- Obtain the MAP estimate $\overline{t}$;
- (b)
- Define ${y}^{1}={y}_{1,\dots ,\overline{t}}$ and ${y}^{2}={y}_{\overline{t}+1,\dots ,N}$;
- (c)
- (Stopping criterion)
**If**$var({y}^{1})=var({y}^{2})$, stop, returning the empty vector $t=\left[\right]$; - (d)
**Else**return the concatenated vector $[seg({y}^{1},{n}_{min})\phantom{\rule{1.em}{0ex}}\overline{t}\phantom{\rule{1.em}{0ex}}\overline{t}+seg({y}^{2},{n}_{min})]$.

**seg**appearing in the last line refers to the function itself; our algorithm, then, is of a recursive nature.

- Obtain ${s}_{0}=su{p}_{{\Theta}_{0}}P({\sigma}_{0},\delta |y)$;
- Obtain the evidence favoring ${H}_{0}$ : $ev({H}_{0},y)=1-{\int}_{T(y)}P({\sigma}_{0},\delta |y)d{\sigma}_{0}d\delta $;
**If**$ev({H}_{0},y)<{\alpha}_{min}$, return 1 ($var({y}^{1})\ne var({y}^{2})$);**else**return 0.

## 4. Results: Simulated Signal

^{®}with little concern for computational performance. In particular, no parallelization strategy was adopted, and both steps are strongly parallelizable. With this issue in mind, we are working on a new version of the algorithm implemented in Python and parallelized using the multiprocessing library; we expect this new version to sensibly improve the computational performance of our algorithm.

## 5. Results: OceanPod Samples

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Algorithm | Parameters | Elapsed Time (s) | # of Segments | First Cutting Point | Last Cutting Point |
---|---|---|---|---|---|

SeqSeg | $\beta =1,\alpha =0.01$ | 9.8912 | 5 | 4990 | 15,001 |

SeqSeg | $\beta =0.01,\alpha =0.01$ | 11.8567 | 5 | 4990 | 15,001 |

SeqSeg | $\beta =1,\alpha =0.1$ | 12.7605 | 5 | 4990 | 15,001 |

SeqSeg | $\beta =0.01,\alpha =0.1$ | 11.5746 | 5 | 4990 | 15,001 |

Palshikar ${S}_{1}$ | $h=3,k=100$ | 0.3116 | 31 | 4778 | 14,852 |

Palshikar ${S}_{2}$ | $h=3,k=100$ | 0.3303 | 13 | 5039 | 14,470 |

Palshikar ${S}_{3}$ | $h=3,k=100$ | 0.3443 | 9 | 1608 | 14,249 |

Palshikar ${S}_{1}$ | $h=3,k=500$ | 0.4264 | 5 | 5039 | 14,179 |

Palshikar ${S}_{2}$ | $h=3,k=500$ | 1.4768 | 55 | 1608 | 19,892 |

Palshikar ${S}_{3}$ | $h=3,k=500$ | 1.3090 | 25 | 964 | 19,892 |

Palshikar ${S}_{1}$ | $h=3,k=1000$ | 1.0846 | 16 | 964 | 19,892 |

Palshikar ${S}_{2}$ | $h=3,k=1000$ | 1.2406 | 10 | 964 | 19,892 |

Palshikar ${S}_{3}$ | $h=3,k=1000$ | 1.4050 | 55 | 1608 | 19,892 |

Palshikar ${S}_{1}$ | $h=3,k=2000$ | 0.9451 | 25 | 964 | 19,892 |

Palshikar ${S}_{2}$ | $h=3,k=2000$ | 0.9889 | 16 | 964 | 19,892 |

Palshikar ${S}_{3}$ | $h=3,k=2000$ | 1.1212 | 10 | 964 | 19,892 |

Palshikar ${S}_{1}$ | $h=4,k=100$ | 0.1718 | 5 | 13,171 | 14,311 |

Palshikar ${S}_{2}$ | $h=4,k=100$ | 0.2303 | 3 | 13,171 | 14,311 |

Palshikar ${S}_{3}$ | $h=4,k=100$ | 0.3023 | 4 | 12,606 | 14,727 |

Palshikar ${S}_{1}$ | $h=4,k=500$ | 0.4439 | 3 | 12,606 | 14,727 |

Palshikar ${S}_{2}$ | $h=4,k=500$ | 0.8924 | 16 | 1608 | 14,923 |

Palshikar ${S}_{3}$ | $h=4,k=500$ | 0.9922 | 10 | 1608 | 14,923 |

Palshikar ${S}_{1}$ | $h=4,k=1000$ | 1.0793 | 9 | 1608 | 14,727 |

Palshikar ${S}_{2}$ | $h=4,k=1000$ | 1.2486 | 6 | 1608 | 14,311 |

Palshikar ${S}_{3}$ | $h=4,k=1000$ | 0.8439 | 16 | 1608 | 14,923 |

Palshikar ${S}_{1}$ | $h=4,k=2000$ | 0.9167 | 10 | 1608 | 14,923 |

Palshikar ${S}_{2}$ | $h=4,k=2000$ | 1.0030 | 9 | 1608 | 14,727 |

Palshikar ${S}_{3}$ | $h=4,k=2000$ | 1.1367 | 6 | 1608 | 14,311 |

Algorithm | Sample | Elapsed Time (s) | Segments | First Segment (min) | Last Segment (min) |
---|---|---|---|---|---|

SeqSeg, $\beta =1{e}^{-5}$ | 8 February 2015 | 245.41 | 28 | 1.80 | 13.32 |

SeqSeg, $\beta =3{e}^{-5}$ | 8 February 2015 | 174.13 | 13 | 1.80 | 13.32 |

Palshikar $h=3$ | 8 February 2015 | 113.33 | 56 | 0.51 | 13.30 |

Palshikar $h=5$ | 8 February 2015 | 112.23 | 29 | 6.65 | 13.28 |

SeqSeg, $\beta =1{e}^{-5}$ | 30 January 2015 | 149.61 | 3 | 2.02 | 10.63 |

SeqSeg, $\beta =3{e}^{-5}$ | 30 January 2015 | 44.83 | 0 | - | - |

Palshikar $h=3$ | 30 January 2015 | 88.27 | 130 | 0.00 | 14.91 |

Palshikar $h=5$ | 30 January 2015 | 87.30 | 27 | 0.30 | 14.21 |

SeqSeg, $\beta =1{e}^{-5}$ | 2 February 2015 | 101.45 | 7 | 1.77 | 10.90 |

SeqSeg, $\beta =3{e}^{-5}$ | 2 February 2015 | 94.24 | 3 | 3.75 | 10.32 |

Palshikar $h=3$ | 2 February 2015 | 89.07 | 114 | 0.00 | 14.83 |

Palshikar $h=5$ | 2 February 2015 | 91.15 | 22 | 0.69 | 14.20 |

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Hubert, P.; Padovese, L.; Stern, J.M. A Sequential Algorithm for Signal Segmentation. *Entropy* **2018**, *20*, 55.
https://doi.org/10.3390/e20010055

**AMA Style**

Hubert P, Padovese L, Stern JM. A Sequential Algorithm for Signal Segmentation. *Entropy*. 2018; 20(1):55.
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**Chicago/Turabian Style**

Hubert, Paulo, Linilson Padovese, and Julio Michael Stern. 2018. "A Sequential Algorithm for Signal Segmentation" *Entropy* 20, no. 1: 55.
https://doi.org/10.3390/e20010055